2021
DOI: 10.1186/s13662-020-03105-x
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Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes

Abstract: In this paper, we prove some basic properties of the discrete Muckenhoupt class $\mathcal{A}^{p}$ A p and the discrete Gehring class $\mathcal{G}^{q}$ G q . These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes.

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Cited by 8 publications
(6 citation statements)
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“…These results further illustrate the value of the extrapolation of families of pairs of functions. To prove these results we need the following inclusion properties of Muckenhoupt classes that has been proved in [28].…”
Section: Apmentioning
confidence: 99%
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“…These results further illustrate the value of the extrapolation of families of pairs of functions. To prove these results we need the following inclusion properties of Muckenhoupt classes that has been proved in [28].…”
Section: Apmentioning
confidence: 99%
“…Lemma 1.1. [28] Assume that w is a nonnegative weight and 1 < p, q < ∞ are positive real numbers. Then the following properties hold.…”
Section: Introductionmentioning
confidence: 99%
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“…For example, the research on regularity and boundedness of discrete analogues of the corresponding L p operators, as well as the higher summability theorems analogues for higher integrability theorems have been studied by numerous authors (see, e.g., [6,8,16,17,24,28] and references therein). In the articles [23,26] the authors studied the structure and basic properties of the weighted discrete Gehring classes, as well as the relationship between the weighted discrete Gehring and Muckenhoupt classes. In recent years, by utilizing the conformable calculus, many authors proved several results related to some integral inequalities like Chebyshev type inequality [3], Hardy type inequalities [25], Hermite-Hadamard type inequalities [2,12,13], and Iyengar type inequalities [27].…”
Section: Introductionmentioning
confidence: 99%
“…In [32], the authors proved that v ∈ A p , 1 < p < ∞ if and only if v -1 p-1 ∈ A p , where p is the conjugate of p, and if v ∈ A p (1 < p < ∞), then v ∈ A q for every q > p and that v α ∈ A p for any 0 < α < 1. Böttcher and Seybold [4] proved that if v ∈ A p then there is ε = ε p,v such that v r ∈ A p for all r ∈ (1ε, 1 + ε).…”
Section: Introductionmentioning
confidence: 99%